How to compare the cost properly
For one person travelling with a rucksack, the train is usually the cheaper option, and three hours with a change is perfectly tolerable if you are not carrying much.
What changes the comparison is that a rail or coach fare is charged per person while a taxi fare is charged per car. A Heathrow transfer covers up to four passengers on one quote, so four travelling together are effectively splitting the fare four ways — before anyone has bought a ticket across London. Where that lands against rail depends entirely on the fares available for your date.
So do the sum for your own trip rather than taking a rule of thumb: look up the rail or coach fare for your date and time, multiply by the number of travelling, add the cross-London leg and anything you would spend getting to Brockenhurst station, then compare that with a written quote. It takes a few minutes and it is the only comparison that reflects what you would actually pay.